The standard model of particle physics (SM) does not account for gravitational interactions.
A number of theoretical models beyond the SM
naturally include gravity and require the existence
of extra dimensions where only gravity propagates freely.
In particular, a model proposed in Ref. [1]}
assumes there are large extra dimensions (LED) compactified on a sub-millimeter scale \(R\) .
In this model gravity can become strong at the \(\mathrm {Te\hspace{-1.00006pt}V}\)  scale,
which effectively solves the hierarchy problem present in the SM.
For \(n\)  extra dimensions
the four-dimensional Planck mass \(M_{\mathrm {Pl}}\)  is related to the \(4+n\) -dimensional
fundamental scale \(M_D\)  by \(M_{\mathrm {Pl}}^2 = 8\pi M_D^2 (M_DR)^n\) .
For \(M_D\)  in the \(\mathrm {Te\hspace{-1.00006pt}V}\)  range \(R\)  is of the order of 0.1 \(\mathrm {mm}\)  (10 \(\mathrm {fm}\) ) for \(n=2\)  (6).
To solve the hierarchy problem \(M_D\)  should not be too large, which rules out \(n=1\) .
Tests of the Newton's law constrain \(R\)  to be below \(37~\mathrm {\mu m}\)  for \(n=2\)  [2]}.
Other constraints come from astrophysics (up to \(n=3\) )
and from high energy colliders [2]}.
